Exercise 8, § 5

Tag 03MK, permanent. Cite it as [Bourbaki, Tag 03MK].

Theory of Sets, III, 237

Let $p_n$ be the number of permutations $u$ of a set E with $n$ elements such that $u(x) \neq x$ for all $x \in \mathrm{E}$ . Show that

p_n = n! - \binom{n}{1}(n-1)! + \binom{n}{2}(n-2)! - \cdots + (-1)^n

\* and hence that $p_n \sim n!/e$ as $n \to +\infty$ \* (same method as in Exercise 7 (a)).

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